Here's a simple script to retrieve the approximate number of backlinks Yahoo has for a site. The function and how to call it follows:

function get_yahoo_backlinks_count($url) {
	$appid = ''; // get this from https://developer.apps.yahoo.com/wsregapp/
	$url = "http://search.yahooapis.com/WebSearchService/V1/webSearch?appid=$appid&query=site:$url&results=1";

	$ch = curl_init();
	curl_setopt($ch, CURLOPT_URL, $url);
	curl_setopt($ch, CURLOPT_RETURNTRANSFER, 1);
	curl_setopt($ch, CURLOPT_REFERER, 'http://www.yoursite.com/');
	$response = curl_exec($ch);
	curl_close($ch);

	$xml = simplexml_load_string($response);
	return $xml -> attributes() -> totalResultsAvailable;
}

echo get_yahoo_backlinks_count('google.com');